Hydrodynamic modeling

In modeling an RO unit, two aspects should be considered membrane transport equations and hydrodynamic modeling of the RO module. The membrane transport equations represent the phenomena (water permeation, solute flux, etc.) taking place at the membrane surface. On the other hand, the hydrodynamic model deals with the macroscopic transport of the various species along with the momentum and energy associated with them. In recent years, a number of mathematical... 

The highly detailed results obtained for the neat ionic liquid [BMIM][PFg] clearly demonstrate the potential of this method for determination of molecular reorienta-tional dynamics in ionic liquids. Further studies should combine the results for the reorientational dynamics with viscosity data in order to compare experimental correlation times with correlation times calculated from hydrodynamic models (cf [14]). It should thus be possible to draw conclusions about the intermolecular structure and interactions in ionic liquids and about the molecular basis of specific properties of ionic liquids. 

It is even more difficult, relying on hydrodynamical models, to explain the frequently observed decrease, rather than increase, of viscosity when a certain quantity of filler is added [47-50]. 

Although the derived scaling law seems to conform to the experimental results under 0-conditions where v = 0.50, the slender body hydrodynamics model is unsatisfactory in many respects. 

In the absence of diffusion, all hydrodynamic models show infinite variances. This is a consequence of the zero-slip condition of hydrodynamics that forces Vz = 0 at the walls of a vessel. In real systems, molecular diffusion will ultimately remove molecules from the stagnant regions near walls. For real systems, W t) will asymptotically approach an exponential distribution and will have finite moments of all orders. However, molecular diffusivities are low for liquids, and may be large indeed. This fact suggests the general inappropriateness of using to characterize the residence time distribution in a laminar flow system. Turbulent flow is less of a problem due to eddy diffusion that typically results in an exponentially decreasing tail at fairly low multiples of the mean residence time. 

Micromixing Models. Hydrodynamic models have intrinsic levels of micromixing. Examples include laminar flow with or without diffusion and the axial dispersion model. Predictions from such models are used directly without explicit concern for micromixing. The residence time distribution corresponding to the models could be associated with a range of micromixing, but this would be inconsistent with the physical model. 

Cho, C. H., Park, S. S., andAhn, Y, Three-Dimensional Wafer Scale Hydrodynamic Modeling for Chemical Mechanical Polishing," Thin Solid Films, Vol. 389,2001, pp. 254-260. 

Figure 3.53. Hydrodynamic model representation of an extraction column.

An alternative description of membrane stability has been based on hydrodynamic models, originally developed for liquid films in various environments [54-56]. Rupture of the film was rationalized by the instability of symmetrical squeezing modes (SQM) related to the thickness fluctuations. In the simplest form it can be described by a condition [54] d Vdis/dh < where is the interaction contribution related to the dis-... 

Urrs, ULTB> aLs and otTB can be found from the hydrodynamic model, and CfTP is calculated in the same manner as discussed previously for bubbly flow. 

Hancox, W. T., and W. B. Nicoll, 1971, A General Technique for the Prediction of Void Distributions in Non-Steady Two-Phase Forced Convection, Int. J. Heat Mass Transfer 14 1377 1394. (3) Haramura, Y., and Y. Katto, 1983, A New Hydrodynamic Model of CHF Applicable Widely to Both Pool and Forced Convection Boiling on Submerged Bodies in Saturated Liquids, Int. J. Heat Mass Transfer 26(3) 387 399. (2)... 

Loutfy and coworkers [29, 30] assumed a different mechanism of interaction between the molecular rotor molecule and the surrounding solvent. The basic assumption was a proportionality of the diffusion constant D of the rotor in a solvent and the rotational reorientation rate kOI. Deviations from the Debye-Stokes-Einstein hydrodynamic model were observed, and Loutfy and Arnold [29] found that the reorientation rate followed a behavior analogous to the Gierer-Wirtz model [31]. The Gierer-Wirtz model considers molecular free volume and leads to a power-law relationship between the reorientation rate and viscosity. The molecular free volume can be envisioned as the void space between the packed solvent molecules, and Doolittle found an empirical relationship between free volume and viscosity [32] (6),... 

The high temperatures and pressures created during transient cavitation are difficult both to calculate and to determine experimentally. The simplest models of collapse, which neglect heat transport and the effects of condensable vapor, predict maximum temperatures and pressures as high as 10,000 K and 10,000 atmospheres. More realistic estimates from increasingly sophisticated hydrodynamic models yield estimates of 5000 K and 1000 atmospheres with effective residence times of <100 nseconds, but the models are very sensitive to initial assumptions of the boundary conditions (30-32). 

The effect of the bulk solution temperature lies primarily in its influence on the bubble content before collapse. With increasing temperature, in general, sonochemical reaction rates are slower. This reflects the dramatic influence which solvent vapor pressure has on the cavitation event the greater the solvent vapor pressure found within a bubble prior to collapse, the less effective the collapse. In fact, one can quantitate this relationship rather well (89). From simple hydrodynamic models of the cavitation process, Neppiras, for example, derives (26) the peak temperature generated during collapse of a gas-filled cavity as... 

Schouten, J. C., and VandenBleek, C. M., Chaotic Behavior in the Hydrodynamic Model of a Fluidized Bed Reactor, Proc. llthlnt. Conf. on Fluidized Bed Comb., 1 459(1991)... 

The flow field of the impacting droplet and its surrounding gas is simulated using a finite-volume solution of the governing equations in a 3-D Cartesian coordinate system. The level-set method is employed to simulate the movement and deformation of the free surface of the droplet during impact. The details of the hydrodynamic model and the numerical scheme are described in Sections... 

There are basically three types of approaches to define the solid stress tensor, or more specifically the solid viscosity. In the early hydrodynamic models— developed by Jackson and his co-workers (Anderson and Jackson, 1967 Anderson et al., 1995), Kuipers et al., (1992), and Tsuo and Gidaspow (1990)—the viscosity is defined as an empirical constant, and also the dependence of the solid phase pressure on the solid volume fraction is determined from experiments. The advantage of this model is its simplicity, the drawback is that it does not take into account the underlying characteristics of the solid phase rheology. 

Kuipers, J. A. M., Hoomans, B. P. B, and van Swaaij, W. P. M. Hydrodynamic Modeling of Gas-Fluidized Beds and their Role for Design and Operation of Fluidized Bed Chemical Reactors. Proceedings of the Fluidization IX conference, 15-30, Durango, USA (1998). 

Tragardh, Ch., A Hydrodynamic Model for the Simulation of an Aerated Agitated Fed-Batch Fermentor . Proceedings of the 2nd International Conference on Bioreactor Fluid Dynamics, Cambridge, UK 117-134 (1988). 

Three primary problem areas exist in dating groundwater. These are (1) Formulation of realistic geochemical-hydrodynamic models needed to interpret data which are generated by field and laboratory measurements, (2) development of sensitive and accurate analytical methods needed to measure trace amounts of various stable and unstable nuclides, and (3) theoretical and field oriented studies to determine with greater accuracy the extent and distribution of the subsurface production of radionuclides which are commonly assumed to originate only in the atmosphere. 

After introducing some types of moving-particle reactors, their advantages and disadvantages, and examples of reactions conducted in them, we consider particular design features. These relate to fluid-particle interactions (extension of the treatment in Chapter 21) and to the complex flow pattern of fluid and solid particles. The latter requires development of a hydrodynamic model as a precursor to a reactor model. We describe these in detail only for particular types of fluidized-bed reactors. 

A hydrodynamic model of fluidization attempts to account for several essential features of fluidization mixing and distribution of solids and fluid in a so-called emulsion region, the formation and motion of bubbles through the bed (the bubble region ), the nature of the bubbles (including their size) and how they affect particle motion/distribution, and the exchange of material between the bubbles (with little solid content) and the predominantly solid emulsion. Models fall into one of three classes (Yates, 1983, pp. 74-78) ... 

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